We develop a general axiomatic theory of algebraic pairs, which simultaneously generalizes several algebraic structures, in order to bypass negation as much as feasible. We investigate several classical theorems and notions in this setting including fractions, integral extensions, and Hilbert's Nullstellensatz. Finally, we study a notion of growth in this context.
There is an increasing body of evidence that prime near-rings with derivations have ring like behavior, indeed, there are several results (see for example [1], [2], [3], [4], [5] and [8]) asserting that the existence of a suitably-constrained derivation on a prime near-ring forces the near-ring to be a ring. It is our purpose to explore further this ring like behaviour. In this paper we generalize some of the results due to Bell and Mason [4] on near-rings admitting a special type of derivation...
We consider inhomogeneous matrix products over max-plus algebra, where the matrices in the product satisfy certain assumptions under which the matrix products of sufficient length are rank-one, as it was shown in [6] (Shue, Anderson, Dey 1998). We establish a bound on the transient after which any product of matrices whose length exceeds that bound becomes rank-one.
We prove here an Eilenberg type theorem: the so-called conjunctive varieties of rational languages correspond to the pseudovarieties of finite semilattice-ordered monoids. Taking complements of members of a conjunctive variety of languages we get a so-called disjunctive variety. We present here a non-trivial example of such a variety together with an equational characterization of the corresponding pseudovariety.
Let be a strong co-ideal of a commutative semiring with identity. Let be a graph with the set of vertices for some , where two distinct vertices and are adjacent if and only if . We look at the diameter and girth of this graph. Also we discuss when is bipartite. Moreover, studies are done on the planarity, clique, and chromatic number of this graph. Examples illustrating the results are presented.
We show that the validity of Parikh’s theorem for context-free languages depends only on a few equational properties of least pre-fixed points. Moreover, we exhibit an infinite basis of -term equations of continuous commutative idempotent semirings.
We show that the validity of Parikh's theorem for context-free languages depends only on a few equational properties of least pre-fixed points. Moreover, we exhibit an infinite basis of μ-term equations of continuous commutative idempotent semirings.
In this paper, a new kind of graph on a commutative ring is introduced and investigated. Small intersection graph of a ring , denoted by , is a graph with all non-small proper ideals of as vertices and two distinct vertices and are adjacent if and only if is not small in . In this article, some interrelation between the graph theoretic properties of this graph and some algebraic properties of rings are studied. We investigated the basic properties of the small intersection graph as diameter,...
Many infinite finitely generated ideal-simple commutative semirings are additively idempotent. It is not clear whether this is true in general. However, to solve the problem, one can restrict oneself only to parasemifields.
We show in an additive inverse regular semiring with as the set of all multiplicative idempotents and as the set of all additive idempotents, the following conditions are equivalent: (i) For all , implies . (ii) is orthodox. (iii) is a semilattice of groups. This result generalizes the corresponding result of regular ring.